LMFDB - Embedded newform 810.3.d.c.161.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(161,810)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("810.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 $ x^16 - 12x^14 + 138x^12 - 1040x^10 + 5541x^8 - 26220x^6 + 99328x^4 - 202728*x^2 + 181476
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.7
Root			$-1.42311 + 1.82514i$ of defining polynomial
Character	$\chi$	$=$	810.161
Dual form			810.3.d.c.161.11

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +0.993782 q^{7} +2.82843i q^{8} +3.16228 q^{10} -4.98623i q^{11} +1.42026 q^{13} -1.40542i q^{14} +4.00000 q^{16} +24.1344i q^{17} -27.5702 q^{19} -4.47214i q^{20} -7.05159 q^{22} -6.03943i q^{23} -5.00000 q^{25} -2.00856i q^{26} -1.98756 q^{28} -19.1577i q^{29} +27.2044 q^{31} -5.65685i q^{32} +34.1312 q^{34} +2.22216i q^{35} -63.1320 q^{37} +38.9901i q^{38} -6.32456 q^{40} +67.1001i q^{41} -27.6583 q^{43} +9.97245i q^{44} -8.54104 q^{46} +6.75889i q^{47} -48.0124 q^{49} +7.07107i q^{50} -2.84053 q^{52} +87.8018i q^{53} +11.1495 q^{55} +2.81084i q^{56} -27.0930 q^{58} +103.106i q^{59} -72.0128 q^{61} -38.4728i q^{62} -8.00000 q^{64} +3.17581i q^{65} +19.9681 q^{67} -48.2688i q^{68} +3.14262 q^{70} -55.1081i q^{71} +132.751 q^{73} +89.2822i q^{74} +55.1404 q^{76} -4.95522i q^{77} -39.5893 q^{79} +8.94427i q^{80} +94.8939 q^{82} +121.188i q^{83} -53.9661 q^{85} +39.1148i q^{86} +14.1032 q^{88} +74.0513i q^{89} +1.41143 q^{91} +12.0789i q^{92} +9.55851 q^{94} -61.6488i q^{95} +2.28766 q^{97} +67.8998i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 8 q^{7} - 40 q^{13} + 64 q^{16} + 80 q^{19} - 48 q^{22} - 80 q^{25} - 16 q^{28} + 32 q^{31} + 96 q^{34} - 88 q^{37} - 184 q^{43} + 24 q^{46} + 168 q^{49} + 80 q^{52} + 152 q^{61} - 128 q^{64}+ \cdots + 32 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 8 * q^7 - 40 * q^13 + 64 * q^16 + 80 * q^19 - 48 * q^22 - 80 * q^25 - 16 * q^28 + 32 * q^31 + 96 * q^34 - 88 * q^37 - 184 * q^43 + 24 * q^46 + 168 * q^49 + 80 * q^52 + 152 * q^61 - 128 * q^64 - 112 * q^67 - 120 * q^70 + 416 * q^73 - 160 * q^76 - 160 * q^79 - 192 * q^82 + 120 * q^85 + 96 * q^88 - 656 * q^91 + 168 * q^94 + 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	0.993782	0.141969	0.0709844	−	0.997477i	$-0.477386\pi$
$7$	0.993782	0.141969	0.0709844	+	0.997477i	$0.477386\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	3.16228	0.316228
$11$	− 4.98623i	− 0.453293i	−0.973977	−	0.226647i	$-0.927224\pi$
$11$	− 4.98623i	− 0.453293i	0.973977	−	0.226647i	$-0.0727763\pi$
$12$	0	0
$13$	1.42026	0.109251	0.0546255	−	0.998507i	$-0.482603\pi$
$13$	1.42026	0.109251	0.0546255	+	0.998507i	$0.482603\pi$
$14$	− 1.40542i	− 0.100387i
$15$	0	0
$16$	4.00000	0.250000
$17$	24.1344i	1.41967i	0.704368	+	0.709835i	$0.251231\pi$
$17$	24.1344i	1.41967i	−0.704368	+	0.709835i	$0.748769\pi$
$18$	0	0
$19$	−27.5702	−1.45106	−0.725531	−	0.688189i	$-0.758406\pi$
$19$	−27.5702	−1.45106	−0.725531	+	0.688189i	$0.758406\pi$
$20$	− 4.47214i	− 0.223607i
$21$	0	0
$22$	−7.05159	−0.320527
$23$	− 6.03943i	− 0.262584i	−0.991344	−	0.131292i	$-0.958087\pi$
$23$	− 6.03943i	− 0.262584i	0.991344	−	0.131292i	$-0.0419125\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	− 2.00856i	− 0.0772522i
$27$	0	0
$28$	−1.98756	−0.0709844
$29$	− 19.1577i	− 0.660609i	−0.943874	−	0.330305i	$-0.892849\pi$
$29$	− 19.1577i	− 0.660609i	0.943874	−	0.330305i	$-0.107151\pi$
$30$	0	0
$31$	27.2044	0.877561	0.438780	−	0.898594i	$-0.355411\pi$
$31$	27.2044	0.877561	0.438780	+	0.898594i	$0.355411\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	34.1312	1.00386
$35$	2.22216i	0.0634904i
$36$	0	0
$37$	−63.1320	−1.70627	−0.853136	−	0.521689i	$-0.825302\pi$
$37$	−63.1320	−1.70627	−0.853136	+	0.521689i	$0.825302\pi$
$38$	38.9901i	1.02606i
$39$	0	0
$40$	−6.32456	−0.158114
$41$	67.1001i	1.63659i	0.574799	+	0.818294i	$0.305080\pi$
$41$	67.1001i	1.63659i	−0.574799	+	0.818294i	$0.694920\pi$
$42$	0	0
$43$	−27.6583	−0.643217	−0.321609	−	0.946873i	$-0.604224\pi$
$43$	−27.6583	−0.643217	−0.321609	+	0.946873i	$0.604224\pi$
$44$	9.97245i	0.226647i
$45$	0	0
$46$	−8.54104	−0.185675
$47$	6.75889i	0.143806i	0.997412	+	0.0719031i	$0.0229072\pi$
$47$	6.75889i	0.143806i	−0.997412	+	0.0719031i	$0.977093\pi$
$48$	0	0
$49$	−48.0124	−0.979845
$50$	7.07107i	0.141421i
$51$	0	0
$52$	−2.84053	−0.0546255
$53$	87.8018i	1.65664i	0.560257	+	0.828319i	$0.310702\pi$
$53$	87.8018i	1.65664i	−0.560257	+	0.828319i	$0.689298\pi$
$54$	0	0
$55$	11.1495	0.202719
$56$	2.81084i	0.0501936i
$57$	0	0
$58$	−27.0930	−0.467121
$59$	103.106i	1.74756i	0.486324	+	0.873778i	$0.338337\pi$
$59$	103.106i	1.74756i	−0.486324	+	0.873778i	$0.661663\pi$
$60$	0	0
$61$	−72.0128	−1.18054	−0.590269	−	0.807207i	$-0.700978\pi$
$61$	−72.0128	−1.18054	−0.590269	+	0.807207i	$0.700978\pi$
$62$	− 38.4728i	− 0.620529i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	3.17581i	0.0488586i
$66$	0	0
$67$	19.9681	0.298031	0.149015	−	0.988835i	$-0.452390\pi$
$67$	19.9681	0.298031	0.149015	+	0.988835i	$0.452390\pi$
$68$	− 48.2688i	− 0.709835i
$69$	0	0
$70$	3.14262	0.0448945
$71$	− 55.1081i	− 0.776170i	−0.921624	−	0.388085i	$-0.873137\pi$
$71$	− 55.1081i	− 0.776170i	0.921624	−	0.388085i	$-0.126863\pi$
$72$	0	0
$73$	132.751	1.81851	0.909254	−	0.416241i	$-0.136653\pi$
$73$	132.751	1.81851	0.909254	+	0.416241i	$0.136653\pi$
$74$	89.2822i	1.20652i
$75$	0	0
$76$	55.1404	0.725531
$77$	− 4.95522i	− 0.0643536i
$78$	0	0
$79$	−39.5893	−0.501130	−0.250565	−	0.968100i	$-0.580616\pi$
$79$	−39.5893	−0.501130	−0.250565	+	0.968100i	$0.580616\pi$
$80$	8.94427i	0.111803i
$81$	0	0
$82$	94.8939	1.15724
$83$	121.188i	1.46009i	0.683398	+	0.730046i	$0.260502\pi$
$83$	121.188i	1.46009i	−0.683398	+	0.730046i	$0.739498\pi$
$84$	0	0
$85$	−53.9661	−0.634896
$86$	39.1148i	0.454823i
$87$	0	0
$88$	14.1032	0.160263
$89$	74.0513i	0.832037i	0.909356	+	0.416018i	$0.136575\pi$
$89$	74.0513i	0.832037i	−0.909356	+	0.416018i	$0.863425\pi$
$90$	0	0
$91$	1.41143	0.0155103
$92$	12.0789i	0.131292i
$93$	0	0
$94$	9.55851	0.101686
$95$	− 61.6488i	− 0.648935i
$96$	0	0
$97$	2.28766	0.0235841	0.0117921	−	0.999930i	$-0.496246\pi$
$97$	2.28766	0.0235841	0.0117921	+	0.999930i	$0.496246\pi$
$98$	67.8998i	0.692855i
$99$	0	0
$100$	10.0000	0.100000
$101$	113.648i	1.12523i	0.826720	+	0.562614i	$0.190204\pi$
$101$	113.648i	1.12523i	−0.826720	+	0.562614i	$0.809796\pi$
$102$	0	0
$103$	−56.7434	−0.550907	−0.275453	−	0.961314i	$-0.588828\pi$
$103$	−56.7434	−0.550907	−0.275453	+	0.961314i	$0.588828\pi$
$104$	4.01711i	0.0386261i
$105$	0	0
$106$	124.170	1.17142
$107$	− 93.1862i	− 0.870899i	−0.900213	−	0.435450i	$-0.856589\pi$
$107$	− 93.1862i	− 0.870899i	0.900213	−	0.435450i	$-0.143411\pi$
$108$	0	0
$109$	−21.7553	−0.199590	−0.0997950	−	0.995008i	$-0.531819\pi$
$109$	−21.7553	−0.199590	−0.0997950	+	0.995008i	$0.531819\pi$
$110$	− 15.7678i	− 0.143344i
$111$	0	0
$112$	3.97513	0.0354922
$113$	− 143.243i	− 1.26763i	−0.773483	−	0.633817i	$-0.781487\pi$
$113$	− 143.243i	− 1.26763i	0.773483	−	0.633817i	$-0.218513\pi$
$114$	0	0
$115$	13.5046	0.117431
$116$	38.3153i	0.330305i
$117$	0	0
$118$	145.814	1.23571
$119$	23.9843i	0.201549i
$120$	0	0
$121$	96.1375	0.794525
$122$	101.842i	0.834767i
$123$	0	0
$124$	−54.4088	−0.438780
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−165.052	−1.29962	−0.649811	−	0.760096i	$-0.725152\pi$
$127$	−165.052	−1.29962	−0.649811	+	0.760096i	$0.725152\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	4.49127	0.0345482
$131$	52.4674i	0.400515i	0.979743	+	0.200257i	$0.0641778\pi$
$131$	52.4674i	0.400515i	−0.979743	+	0.200257i	$0.935822\pi$
$132$	0	0
$133$	−27.3988	−0.206006
$134$	− 28.2391i	− 0.210740i
$135$	0	0
$136$	−68.2624	−0.501929
$137$	− 120.373i	− 0.878636i	−0.898332	−	0.439318i	$-0.855220\pi$
$137$	− 120.373i	− 0.878636i	0.898332	−	0.439318i	$-0.144780\pi$
$138$	0	0
$139$	−66.3984	−0.477687	−0.238843	−	0.971058i	$-0.576768\pi$
$139$	−66.3984	−0.477687	−0.238843	+	0.971058i	$0.576768\pi$
$140$	− 4.44433i	− 0.0317452i
$141$	0	0
$142$	−77.9346	−0.548835
$143$	− 7.08176i	− 0.0495228i
$144$	0	0
$145$	42.8378	0.295433
$146$	− 187.738i	− 1.28588i
$147$	0	0
$148$	126.264	0.853136
$149$	95.6198i	0.641744i	0.947123	+	0.320872i	$0.103976\pi$
$149$	95.6198i	0.641744i	−0.947123	+	0.320872i	$0.896024\pi$
$150$	0	0
$151$	270.524	1.79155	0.895776	−	0.444506i	$-0.146621\pi$
$151$	270.524	1.79155	0.895776	+	0.444506i	$0.146621\pi$
$152$	− 77.9803i	− 0.513028i
$153$	0	0
$154$	−7.00774	−0.0455048
$155$	60.8308i	0.392457i
$156$	0	0
$157$	36.3452	0.231498	0.115749	−	0.993278i	$-0.463073\pi$
$157$	36.3452	0.231498	0.115749	+	0.993278i	$0.463073\pi$
$158$	55.9877i	0.354353i
$159$	0	0
$160$	12.6491	0.0790569
$161$	− 6.00188i	− 0.0372787i
$162$	0	0
$163$	93.3260	0.572552	0.286276	−	0.958147i	$-0.407583\pi$
$163$	93.3260	0.572552	0.286276	+	0.958147i	$0.407583\pi$
$164$	− 134.200i	− 0.818294i
$165$	0	0
$166$	171.385	1.03244
$167$	56.9278i	0.340885i	0.985368	+	0.170443i	$0.0545197\pi$
$167$	56.9278i	0.340885i	−0.985368	+	0.170443i	$0.945480\pi$
$168$	0	0
$169$	−166.983	−0.988064
$170$	76.3197i	0.448939i
$171$	0	0
$172$	55.3167	0.321609
$173$	− 217.640i	− 1.25803i	−0.777391	−	0.629017i	$-0.783458\pi$
$173$	− 217.640i	− 1.25803i	0.777391	−	0.629017i	$-0.216542\pi$
$174$	0	0
$175$	−4.96891	−0.0283938
$176$	− 19.9449i	− 0.113323i
$177$	0	0
$178$	104.724	0.588339
$179$	56.2638i	0.314323i	0.987573	+	0.157161i	$0.0502343\pi$
$179$	56.2638i	0.314323i	−0.987573	+	0.157161i	$0.949766\pi$
$180$	0	0
$181$	−130.959	−0.723528	−0.361764	−	0.932270i	$-0.617825\pi$
$181$	−130.959	−0.723528	−0.361764	+	0.932270i	$0.617825\pi$
$182$	− 1.99607i	− 0.0109674i
$183$	0	0
$184$	17.0821	0.0928374
$185$	− 141.168i	− 0.763068i
$186$	0	0
$187$	120.340	0.643527
$188$	− 13.5178i	− 0.0719031i
$189$	0	0
$190$	−87.1846	−0.458866
$191$	33.9997i	0.178009i	0.996031	+	0.0890045i	$0.0283686\pi$
$191$	33.9997i	0.178009i	−0.996031	+	0.0890045i	$0.971631\pi$
$192$	0	0
$193$	−183.108	−0.948748	−0.474374	−	0.880323i	$-0.657326\pi$
$193$	−183.108	−0.948748	−0.474374	+	0.880323i	$0.657326\pi$
$194$	− 3.23524i	− 0.0166765i
$195$	0	0
$196$	96.0248	0.489922
$197$	30.9351i	0.157031i	0.996913	+	0.0785154i	$0.0250180\pi$
$197$	30.9351i	0.157031i	−0.996913	+	0.0785154i	$0.974982\pi$
$198$	0	0
$199$	−332.873	−1.67273	−0.836363	−	0.548176i	$-0.815322\pi$
$199$	−332.873	−1.67273	−0.836363	+	0.548176i	$0.815322\pi$
$200$	− 14.1421i	− 0.0707107i
$201$	0	0
$202$	160.723	0.795656
$203$	− 19.0385i	− 0.0937860i
$204$	0	0
$205$	−150.040	−0.731905
$206$	80.2473i	0.389550i
$207$	0	0
$208$	5.68105	0.0273128
$209$	137.471i	0.657757i
$210$	0	0
$211$	−160.772	−0.761951	−0.380976	−	0.924585i	$-0.624412\pi$
$211$	−160.772	−0.761951	−0.380976	+	0.924585i	$0.624412\pi$
$212$	− 175.604i	− 0.828319i
$213$	0	0
$214$	−131.785	−0.615819
$215$	− 61.8459i	− 0.287655i
$216$	0	0
$217$	27.0352	0.124586
$218$	30.7667i	0.141131i
$219$	0	0
$220$	−22.2991	−0.101359
$221$	34.2772i	0.155100i
$222$	0	0
$223$	391.401	1.75516	0.877582	−	0.479427i	$-0.159156\pi$
$223$	391.401	1.75516	0.877582	+	0.479427i	$0.159156\pi$
$224$	− 5.62168i	− 0.0250968i
$225$	0	0
$226$	−202.576	−0.896353
$227$	− 340.699i	− 1.50088i	−0.660940	−	0.750439i	$-0.729842\pi$
$227$	− 340.699i	− 1.50088i	0.660940	−	0.750439i	$-0.270158\pi$
$228$	0	0
$229$	−94.4465	−0.412430	−0.206215	−	0.978507i	$-0.566115\pi$
$229$	−94.4465	−0.412430	−0.206215	+	0.978507i	$0.566115\pi$
$230$	− 19.0984i	− 0.0830363i
$231$	0	0
$232$	54.1861	0.233561
$233$	117.982i	0.506362i	0.967419	+	0.253181i	$0.0814769\pi$
$233$	117.982i	0.506362i	−0.967419	+	0.253181i	$0.918523\pi$
$234$	0	0
$235$	−15.1133	−0.0643121
$236$	− 206.212i	− 0.873778i
$237$	0	0
$238$	33.9190	0.142517
$239$	− 369.113i	− 1.54441i	−0.635376	−	0.772203i	$-0.719155\pi$
$239$	− 369.113i	− 1.54441i	0.635376	−	0.772203i	$-0.280845\pi$
$240$	0	0
$241$	−375.630	−1.55863	−0.779316	−	0.626631i	$-0.784433\pi$
$241$	−375.630	−1.55863	−0.779316	+	0.626631i	$0.784433\pi$
$242$	− 135.959i	− 0.561814i
$243$	0	0
$244$	144.026	0.590269
$245$	− 107.359i	− 0.438200i
$246$	0	0
$247$	−39.1569	−0.158530
$248$	76.9456i	0.310265i
$249$	0	0
$250$	−15.8114	−0.0632456
$251$	− 240.639i	− 0.958722i	−0.877618	−	0.479361i	$-0.840869\pi$
$251$	− 240.639i	− 0.958722i	0.877618	−	0.479361i	$-0.159131\pi$
$252$	0	0
$253$	−30.1140	−0.119028
$254$	233.419i	0.918971i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 125.219i	− 0.487233i	−0.969872	−	0.243616i	$-0.921666\pi$
$257$	− 125.219i	− 0.487233i	0.969872	−	0.243616i	$-0.0783338\pi$
$258$	0	0
$259$	−62.7395	−0.242237
$260$	− 6.35161i	− 0.0244293i
$261$	0	0
$262$	74.2001	0.283207
$263$	93.2432i	0.354537i	0.984163	+	0.177269i	$0.0567261\pi$
$263$	93.2432i	0.354537i	−0.984163	+	0.177269i	$0.943274\pi$
$264$	0	0
$265$	−196.331	−0.740871
$266$	38.7477i	0.145668i
$267$	0	0
$268$	−39.9361	−0.149015
$269$	− 76.3366i	− 0.283779i	−0.989882	−	0.141890i	$-0.954682\pi$
$269$	− 76.3366i	− 0.283779i	0.989882	−	0.141890i	$-0.0453178\pi$
$270$	0	0
$271$	529.738	1.95475	0.977377	−	0.211506i	$-0.0678368\pi$
$271$	529.738	1.95475	0.977377	+	0.211506i	$0.0678368\pi$
$272$	96.5376i	0.354918i
$273$	0	0
$274$	−170.233	−0.621290
$275$	24.9311i	0.0906587i
$276$	0	0
$277$	−84.9611	−0.306719	−0.153359	−	0.988170i	$-0.549009\pi$
$277$	−84.9611	−0.306719	−0.153359	+	0.988170i	$0.549009\pi$
$278$	93.9016i	0.337775i
$279$	0	0
$280$	−6.28523	−0.0224473
$281$	214.242i	0.762428i	0.924487	+	0.381214i	$0.124494\pi$
$281$	214.242i	0.762428i	−0.924487	+	0.381214i	$0.875506\pi$
$282$	0	0
$283$	378.158	1.33625	0.668123	−	0.744051i	$-0.267098\pi$
$283$	378.158	1.33625	0.668123	+	0.744051i	$0.267098\pi$
$284$	110.216i	0.388085i
$285$	0	0
$286$	−10.0151	−0.0350179
$287$	66.6829i	0.232345i
$288$	0	0
$289$	−293.469	−1.01546
$290$	− 60.5819i	− 0.208903i
$291$	0	0
$292$	−265.502	−0.909254
$293$	− 50.0000i	− 0.170648i	−0.996353	−	0.0853242i	$-0.972807\pi$
$293$	− 50.0000i	− 0.170648i	0.996353	−	0.0853242i	$-0.0271926\pi$
$294$	0	0
$295$	−230.552	−0.781531
$296$	− 178.564i	− 0.603258i
$297$	0	0
$298$	135.227	0.453781
$299$	− 8.57758i	− 0.0286876i
$300$	0	0
$301$	−27.4864	−0.0913168
$302$	− 382.579i	− 1.26682i
$303$	0	0
$304$	−110.281	−0.362766
$305$	− 161.026i	− 0.527953i
$306$	0	0
$307$	145.083	0.472583	0.236292	−	0.971682i	$-0.424068\pi$
$307$	145.083	0.472583	0.236292	+	0.971682i	$0.424068\pi$
$308$	9.91045i	0.0321768i
$309$	0	0
$310$	86.0278	0.277509
$311$	104.589i	0.336301i	0.985761	+	0.168150i	$0.0537794\pi$
$311$	104.589i	0.336301i	−0.985761	+	0.168150i	$0.946221\pi$
$312$	0	0
$313$	57.7737	0.184581	0.0922903	−	0.995732i	$-0.470581\pi$
$313$	57.7737	0.184581	0.0922903	+	0.995732i	$0.470581\pi$
$314$	− 51.3999i	− 0.163694i
$315$	0	0
$316$	79.1786	0.250565
$317$	− 98.3000i	− 0.310095i	−0.987907	−	0.155047i	$-0.950447\pi$
$317$	− 98.3000i	− 0.310095i	0.987907	−	0.155047i	$-0.0495530\pi$
$318$	0	0
$319$	−95.5245	−0.299450
$320$	− 17.8885i	− 0.0559017i
$321$	0	0
$322$	−8.48794	−0.0263601
$323$	− 665.390i	− 2.06003i
$324$	0	0
$325$	−7.10132	−0.0218502
$326$	− 131.983i	− 0.404856i
$327$	0	0
$328$	−189.788	−0.578622
$329$	6.71686i	0.0204160i
$330$	0	0
$331$	−213.075	−0.643730	−0.321865	−	0.946786i	$-0.604310\pi$
$331$	−213.075	−0.643730	−0.321865	+	0.946786i	$0.604310\pi$
$332$	− 242.375i	− 0.730046i
$333$	0	0
$334$	80.5081	0.241042
$335$	44.6499i	0.133283i
$336$	0	0
$337$	165.031	0.489705	0.244853	−	0.969560i	$-0.421260\pi$
$337$	165.031	0.489705	0.244853	+	0.969560i	$0.421260\pi$
$338$	236.149i	0.698667i
$339$	0	0
$340$	107.932	0.317448
$341$	− 135.647i	− 0.397792i
$342$	0	0
$343$	−96.4092	−0.281076
$344$	− 78.2296i	− 0.227412i
$345$	0	0
$346$	−307.789	−0.889564
$347$	− 90.3578i	− 0.260397i	−0.991488	−	0.130199i	$-0.958439\pi$
$347$	− 90.3578i	− 0.260397i	0.991488	−	0.130199i	$-0.0415615\pi$
$348$	0	0
$349$	−354.975	−1.01712	−0.508561	−	0.861026i	$-0.669822\pi$
$349$	−354.975	−1.01712	−0.508561	+	0.861026i	$0.669822\pi$
$350$	7.02710i	0.0200774i
$351$	0	0
$352$	−28.2064	−0.0801317
$353$	− 361.096i	− 1.02293i	−0.859303	−	0.511467i	$-0.829102\pi$
$353$	− 361.096i	− 1.02293i	0.859303	−	0.511467i	$-0.170898\pi$
$354$	0	0
$355$	123.225	0.347114
$356$	− 148.103i	− 0.416018i
$357$	0	0
$358$	79.5690	0.222260
$359$	229.312i	0.638753i	0.947628	+	0.319377i	$0.103473\pi$
$359$	229.312i	0.638753i	−0.947628	+	0.319377i	$0.896527\pi$
$360$	0	0
$361$	399.115	1.10558
$362$	185.203i	0.511612i
$363$	0	0
$364$	−2.82287	−0.00775513
$365$	296.840i	0.813262i
$366$	0	0
$367$	411.575	1.12146	0.560728	−	0.828000i	$-0.310521\pi$
$367$	411.575	1.12146	0.560728	+	0.828000i	$0.310521\pi$
$368$	− 24.1577i	− 0.0656460i
$369$	0	0
$370$	−199.641	−0.539570
$371$	87.2559i	0.235191i
$372$	0	0
$373$	687.028	1.84190	0.920949	−	0.389684i	$-0.127416\pi$
$373$	687.028	1.84190	0.920949	+	0.389684i	$0.127416\pi$
$374$	− 170.186i	− 0.455042i
$375$	0	0
$376$	−19.1170	−0.0508431
$377$	− 27.2089i	− 0.0721722i
$378$	0	0
$379$	−154.733	−0.408267	−0.204134	−	0.978943i	$-0.565438\pi$
$379$	−154.733	−0.408267	−0.204134	+	0.978943i	$0.565438\pi$
$380$	123.298i	0.324467i
$381$	0	0
$382$	48.0829	0.125871
$383$	163.122i	0.425907i	0.977062	+	0.212954i	$0.0683083\pi$
$383$	163.122i	0.425907i	−0.977062	+	0.212954i	$0.931692\pi$
$384$	0	0
$385$	11.0802	0.0287798
$386$	258.954i	0.670866i
$387$	0	0
$388$	−4.57532	−0.0117921
$389$	− 310.319i	− 0.797734i	−0.917009	−	0.398867i	$-0.869403\pi$
$389$	− 310.319i	− 0.797734i	0.917009	−	0.398867i	$-0.130597\pi$
$390$	0	0
$391$	145.758	0.372783
$392$	− 135.800i	− 0.346427i
$393$	0	0
$394$	43.7488	0.111038
$395$	− 88.5244i	− 0.224112i
$396$	0	0
$397$	−270.057	−0.680244	−0.340122	−	0.940381i	$-0.610468\pi$
$397$	−270.057	−0.680244	−0.340122	+	0.940381i	$0.610468\pi$
$398$	470.753i	1.18280i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	− 606.672i	− 1.51290i	−0.654052	−	0.756449i	$-0.726932\pi$
$401$	− 606.672i	− 1.51290i	0.654052	−	0.756449i	$-0.273068\pi$
$402$	0	0
$403$	38.6374	0.0958744
$404$	− 227.296i	− 0.562614i
$405$	0	0
$406$	−26.9246	−0.0663167
$407$	314.791i	0.773441i
$408$	0	0
$409$	−665.897	−1.62811	−0.814055	−	0.580788i	$-0.802745\pi$
$409$	−665.897	−1.62811	−0.814055	+	0.580788i	$0.802745\pi$
$410$	212.189i	0.517535i
$411$	0	0
$412$	113.487	0.275453
$413$	102.465i	0.248099i
$414$	0	0
$415$	−270.984	−0.652973
$416$	− 8.03422i	− 0.0193130i
$417$	0	0
$418$	194.414	0.465104
$419$	216.212i	0.516020i	0.966142	+	0.258010i	$0.0830667\pi$
$419$	216.212i	0.516020i	−0.966142	+	0.258010i	$0.916933\pi$
$420$	0	0
$421$	315.504	0.749415	0.374708	−	0.927143i	$-0.377743\pi$
$421$	315.504	0.749415	0.374708	+	0.927143i	$0.377743\pi$
$422$	227.366i	0.538781i
$423$	0	0
$424$	−248.341	−0.585710
$425$	− 120.672i	− 0.283934i
$426$	0	0
$427$	−71.5651	−0.167600
$428$	186.372i	0.435450i
$429$	0	0
$430$	−87.4634	−0.203403
$431$	705.119i	1.63601i	0.575213	+	0.818003i	$0.304919\pi$
$431$	705.119i	1.63601i	−0.575213	+	0.818003i	$0.695081\pi$
$432$	0	0
$433$	152.735	0.352737	0.176369	−	0.984324i	$-0.443565\pi$
$433$	152.735	0.352737	0.176369	+	0.984324i	$0.443565\pi$
$434$	− 38.2336i	− 0.0880958i
$435$	0	0
$436$	43.5106	0.0997950
$437$	166.508i	0.381026i
$438$	0	0
$439$	162.235	0.369555	0.184778	−	0.982780i	$-0.440844\pi$
$439$	162.235	0.369555	0.184778	+	0.982780i	$0.440844\pi$
$440$	31.5357i	0.0716720i
$441$	0	0
$442$	48.4753	0.109673
$443$	− 387.565i	− 0.874865i	−0.899251	−	0.437433i	$-0.855888\pi$
$443$	− 387.565i	− 0.874865i	0.899251	−	0.437433i	$-0.144112\pi$
$444$	0	0
$445$	−165.584	−0.372098
$446$	− 553.525i	− 1.24109i
$447$	0	0
$448$	−7.95026	−0.0177461
$449$	111.025i	0.247272i	0.992328	+	0.123636i	$0.0394554\pi$
$449$	111.025i	0.247272i	−0.992328	+	0.123636i	$0.960545\pi$
$450$	0	0
$451$	334.577	0.741855
$452$	286.486i	0.633817i
$453$	0	0
$454$	−481.821	−1.06128
$455$	3.15606i	0.00693640i
$456$	0	0
$457$	−298.799	−0.653828	−0.326914	−	0.945054i	$-0.606009\pi$
$457$	−298.799	−0.653828	−0.326914	+	0.945054i	$0.606009\pi$
$458$	133.567i	0.291632i
$459$	0	0
$460$	−27.0091	−0.0587155
$461$	207.888i	0.450950i	0.974249	+	0.225475i	$0.0723934\pi$
$461$	207.888i	0.450950i	−0.974249	+	0.225475i	$0.927607\pi$
$462$	0	0
$463$	865.125	1.86852	0.934261	−	0.356591i	$-0.116061\pi$
$463$	865.125	1.86852	0.934261	+	0.356591i	$0.116061\pi$
$464$	− 76.6307i	− 0.165152i
$465$	0	0
$466$	166.852	0.358052
$467$	854.726i	1.83025i	0.403172	+	0.915124i	$0.367908\pi$
$467$	854.726i	1.83025i	−0.403172	+	0.915124i	$0.632092\pi$
$468$	0	0
$469$	19.8439	0.0423111
$470$	21.3735i	0.0454755i
$471$	0	0
$472$	−291.627	−0.617855
$473$	137.911i	0.291566i
$474$	0	0
$475$	137.851	0.290212
$476$	− 47.9687i	− 0.100775i
$477$	0	0
$478$	−522.004	−1.09206
$479$	− 304.139i	− 0.634945i	−0.948267	−	0.317473i	$-0.897166\pi$
$479$	− 304.139i	− 0.634945i	0.948267	−	0.317473i	$-0.102834\pi$
$480$	0	0
$481$	−89.6641	−0.186412
$482$	531.222i	1.10212i
$483$	0	0
$484$	−192.275	−0.397263
$485$	5.11537i	0.0105472i
$486$	0	0
$487$	907.378	1.86320	0.931600	−	0.363486i	$-0.118413\pi$
$487$	907.378	1.86320	0.931600	+	0.363486i	$0.118413\pi$
$488$	− 203.683i	− 0.417383i
$489$	0	0
$490$	−151.829	−0.309854
$491$	65.4467i	0.133293i	0.997777	+	0.0666464i	$0.0212299\pi$
$491$	65.4467i	0.133293i	−0.997777	+	0.0666464i	$0.978770\pi$
$492$	0	0
$493$	462.359	0.937847
$494$	55.3763i	0.112098i
$495$	0	0
$496$	108.818	0.219390
$497$	− 54.7654i	− 0.110192i
$498$	0	0
$499$	717.284	1.43744	0.718721	−	0.695298i	$-0.244728\pi$
$499$	717.284	1.43744	0.718721	+	0.695298i	$0.244728\pi$
$500$	22.3607i	0.0447214i
$501$	0	0
$502$	−340.315	−0.677919
$503$	− 623.374i	− 1.23931i	−0.784873	−	0.619657i	$-0.787272\pi$
$503$	− 623.374i	− 1.23931i	0.784873	−	0.619657i	$-0.212728\pi$
$504$	0	0
$505$	−254.125	−0.503217
$506$	42.5876i	0.0841652i
$507$	0	0
$508$	330.104	0.649811
$509$	666.448i	1.30933i	0.755920	+	0.654664i	$0.227190\pi$
$509$	666.448i	1.30933i	−0.755920	+	0.654664i	$0.772810\pi$
$510$	0	0
$511$	131.926	0.258172
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−177.086	−0.344526
$515$	− 126.882i	− 0.246373i
$516$	0	0
$517$	33.7013	0.0651864
$518$	88.7270i	0.171288i
$519$	0	0
$520$	−8.98254	−0.0172741
$521$	896.544i	1.72081i	0.509608	+	0.860407i	$0.329790\pi$
$521$	896.544i	1.72081i	−0.509608	+	0.860407i	$0.670210\pi$
$522$	0	0
$523$	354.070	0.676999	0.338499	−	0.940967i	$-0.390081\pi$
$523$	354.070	0.676999	0.338499	+	0.940967i	$0.390081\pi$
$524$	− 104.935i	− 0.200257i
$525$	0	0
$526$	131.866	0.250696
$527$	656.561i	1.24585i
$528$	0	0
$529$	492.525	0.931050
$530$	277.654i	0.523875i
$531$	0	0
$532$	54.7975	0.103003
$533$	95.2999i	0.178799i
$534$	0	0
$535$	208.371	0.389478
$536$	56.4782i	0.105370i
$537$	0	0
$538$	−107.956	−0.200662
$539$	239.401i	0.444157i
$540$	0	0
$541$	−392.737	−0.725947	−0.362974	−	0.931799i	$-0.618238\pi$
$541$	−392.737	−0.725947	−0.362974	+	0.931799i	$0.618238\pi$
$542$	− 749.163i	− 1.38222i
$543$	0	0
$544$	136.525	0.250965
$545$	− 48.6463i	− 0.0892594i
$546$	0	0
$547$	271.057	0.495534	0.247767	−	0.968820i	$-0.420303\pi$
$547$	271.057	0.495534	0.247767	+	0.968820i	$0.420303\pi$
$548$	240.746i	0.439318i
$549$	0	0
$550$	35.2579	0.0641054
$551$	528.180i	0.958585i
$552$	0	0
$553$	−39.3431	−0.0711449
$554$	120.153i	0.216883i
$555$	0	0
$556$	132.797	0.238843
$557$	− 122.363i	− 0.219682i	−0.993949	−	0.109841i	$-0.964966\pi$
$557$	− 122.363i	− 0.219682i	0.993949	−	0.109841i	$-0.0350342\pi$
$558$	0	0
$559$	−39.2821	−0.0702722
$560$	8.88866i	0.0158726i
$561$	0	0
$562$	302.984	0.539118
$563$	52.9120i	0.0939823i	0.998895	+	0.0469911i	$0.0149633\pi$
$563$	52.9120i	0.0939823i	−0.998895	+	0.0469911i	$0.985037\pi$
$564$	0	0
$565$	320.301	0.566904
$566$	− 534.796i	− 0.944869i
$567$	0	0
$568$	155.869	0.274418
$569$	729.154i	1.28147i	0.767764	+	0.640733i	$0.221369\pi$
$569$	729.154i	1.28147i	−0.767764	+	0.640733i	$0.778631\pi$
$570$	0	0
$571$	−397.481	−0.696115	−0.348057	−	0.937473i	$-0.613159\pi$
$571$	−397.481	−0.696115	−0.348057	+	0.937473i	$0.613159\pi$
$572$	14.1635i	0.0247614i
$573$	0	0
$574$	94.3039	0.164293
$575$	30.1971i	0.0525168i
$576$	0	0
$577$	−82.2138	−0.142485	−0.0712425	−	0.997459i	$-0.522696\pi$
$577$	−82.2138	−0.142485	−0.0712425	+	0.997459i	$0.522696\pi$
$578$	415.028i	0.718041i
$579$	0	0
$580$	−85.6757	−0.147717
$581$	120.434i	0.207288i
$582$	0	0
$583$	437.800	0.750943
$584$	375.477i	0.642940i
$585$	0	0
$586$	−70.7106	−0.120667
$587$	861.368i	1.46741i	0.679470	+	0.733703i	$0.262210\pi$
$587$	861.368i	1.46741i	−0.679470	+	0.733703i	$0.737790\pi$
$588$	0	0
$589$	−750.030	−1.27340
$590$	326.049i	0.552626i
$591$	0	0
$592$	−252.528	−0.426568
$593$	377.445i	0.636500i	0.948007	+	0.318250i	$0.103095\pi$
$593$	377.445i	0.636500i	−0.948007	+	0.318250i	$0.896905\pi$
$594$	0	0
$595$	−53.6306	−0.0901355
$596$	− 191.240i	− 0.320872i
$597$	0	0
$598$	−12.1305	−0.0202852
$599$	− 309.928i	− 0.517408i	−0.965957	−	0.258704i	$-0.916705\pi$
$599$	− 309.928i	− 0.517408i	0.965957	−	0.258704i	$-0.0832955\pi$
$600$	0	0
$601$	417.783	0.695146	0.347573	−	0.937653i	$-0.387006\pi$
$601$	417.783	0.695146	0.347573	+	0.937653i	$0.387006\pi$
$602$	38.8716i	0.0645708i
$603$	0	0
$604$	−541.049	−0.895776
$605$	214.970i	0.355322i
$606$	0	0
$607$	856.166	1.41049	0.705244	−	0.708965i	$-0.250838\pi$
$607$	856.166	1.41049	0.705244	+	0.708965i	$0.250838\pi$
$608$	155.961i	0.256514i
$609$	0	0
$610$	−227.725	−0.373319
$611$	9.59940i	0.0157110i
$612$	0	0
$613$	−1022.92	−1.66870	−0.834352	−	0.551231i	$-0.814158\pi$
$613$	−1022.92	−1.66870	−0.834352	+	0.551231i	$0.814158\pi$
$614$	− 205.178i	− 0.334167i
$615$	0	0
$616$	14.0155	0.0227524
$617$	− 585.968i	− 0.949704i	−0.880066	−	0.474852i	$-0.842502\pi$
$617$	− 585.968i	− 0.949704i	0.880066	−	0.474852i	$-0.157498\pi$
$618$	0	0
$619$	−911.575	−1.47266	−0.736329	−	0.676624i	$-0.763442\pi$
$619$	−911.575	−1.47266	−0.736329	+	0.676624i	$0.763442\pi$
$620$	− 121.662i	− 0.196229i
$621$	0	0
$622$	147.912	0.237800
$623$	73.5909i	0.118123i
$624$	0	0
$625$	25.0000	0.0400000
$626$	− 81.7044i	− 0.130518i
$627$	0	0
$628$	−72.6905	−0.115749
$629$	− 1523.65i	− 2.42234i
$630$	0	0
$631$	463.111	0.733932	0.366966	−	0.930234i	$-0.380396\pi$
$631$	463.111	0.733932	0.366966	+	0.930234i	$0.380396\pi$
$632$	− 111.975i	− 0.177176i
$633$	0	0
$634$	−139.017	−0.219270
$635$	− 369.067i	− 0.581208i
$636$	0	0
$637$	−68.1903	−0.107049
$638$	135.092i	0.211743i
$639$	0	0
$640$	−25.2982	−0.0395285
$641$	591.726i	0.923129i	0.887107	+	0.461565i	$0.152712\pi$
$641$	591.726i	0.923129i	−0.887107	+	0.461565i	$0.847288\pi$
$642$	0	0
$643$	−747.102	−1.16190	−0.580950	−	0.813939i	$-0.697319\pi$
$643$	−747.102	−1.16190	−0.580950	+	0.813939i	$0.697319\pi$
$644$	12.0038i	0.0186394i
$645$	0	0
$646$	−941.003	−1.45666
$647$	86.2103i	0.133246i	0.997778	+	0.0666231i	$0.0212225\pi$
$647$	86.2103i	0.133246i	−0.997778	+	0.0666231i	$0.978777\pi$
$648$	0	0
$649$	514.109	0.792156
$650$	10.0428i	0.0154504i
$651$	0	0
$652$	−186.652	−0.286276
$653$	− 171.547i	− 0.262707i	−0.991336	−	0.131353i	$-0.958068\pi$
$653$	− 171.547i	− 0.262707i	0.991336	−	0.131353i	$-0.0419322\pi$
$654$	0	0
$655$	−117.321	−0.179116
$656$	268.401i	0.409147i
$657$	0	0
$658$	9.49908	0.0144363
$659$	− 94.2318i	− 0.142992i	−0.997441	−	0.0714960i	$-0.977223\pi$
$659$	− 94.2318i	− 0.142992i	0.997441	−	0.0714960i	$-0.0227773\pi$
$660$	0	0
$661$	844.781	1.27803	0.639017	−	0.769192i	$-0.279341\pi$
$661$	844.781	1.27803	0.639017	+	0.769192i	$0.279341\pi$
$662$	301.333i	0.455186i
$663$	0	0
$664$	−342.771	−0.516221
$665$	− 61.2655i	− 0.0921286i
$666$	0	0
$667$	−115.701	−0.173465
$668$	− 113.856i	− 0.170443i
$669$	0	0
$670$	63.1446	0.0942456
$671$	359.072i	0.535130i
$672$	0	0
$673$	−704.290	−1.04649	−0.523247	−	0.852181i	$-0.675279\pi$
$673$	−704.290	−1.04649	−0.523247	+	0.852181i	$0.675279\pi$
$674$	− 233.389i	− 0.346274i
$675$	0	0
$676$	333.966	0.494032
$677$	1278.87i	1.88903i	0.328475	+	0.944513i	$0.393465\pi$
$677$	1278.87i	1.88903i	−0.328475	+	0.944513i	$0.606535\pi$
$678$	0	0
$679$	2.27344	0.00334822
$680$	− 152.639i	− 0.224470i
$681$	0	0
$682$	−191.834	−0.281282
$683$	− 571.914i	− 0.837356i	−0.908135	−	0.418678i	$-0.862494\pi$
$683$	− 571.914i	− 0.837356i	0.908135	−	0.418678i	$-0.137506\pi$
$684$	0	0
$685$	269.163	0.392938
$686$	136.343i	0.198751i
$687$	0	0
$688$	−110.633	−0.160804
$689$	124.702i	0.180989i
$690$	0	0
$691$	620.122	0.897427	0.448713	−	0.893676i	$-0.351882\pi$
$691$	620.122	0.897427	0.448713	+	0.893676i	$0.351882\pi$
$692$	435.280i	0.629017i
$693$	0	0
$694$	−127.785	−0.184129
$695$	− 148.471i	− 0.213628i
$696$	0	0
$697$	−1619.42	−2.32342
$698$	502.011i	0.719213i
$699$	0	0
$700$	9.93782	0.0141969
$701$	− 66.6004i	− 0.0950077i	−0.998871	−	0.0475038i	$-0.984873\pi$
$701$	− 66.6004i	− 0.0950077i	0.998871	−	0.0475038i	$-0.0151266\pi$
$702$	0	0
$703$	1740.56	2.47591
$704$	39.8898i	0.0566617i
$705$	0	0
$706$	−510.666	−0.723324
$707$	112.941i	0.159747i
$708$	0	0
$709$	−210.574	−0.297002	−0.148501	−	0.988912i	$-0.547445\pi$
$709$	−210.574	−0.297002	−0.148501	+	0.988912i	$0.547445\pi$
$710$	− 174.267i	− 0.245447i
$711$	0	0
$712$	−209.449	−0.294169
$713$	− 164.299i	− 0.230433i
$714$	0	0
$715$	15.8353	0.0221473
$716$	− 112.528i	− 0.157161i
$717$	0	0
$718$	324.297	0.451667
$719$	900.163i	1.25197i	0.779837	+	0.625983i	$0.215302\pi$
$719$	900.163i	1.25197i	−0.779837	+	0.625983i	$0.784698\pi$
$720$	0	0
$721$	−56.3906	−0.0782116
$722$	− 564.434i	− 0.781764i
$723$	0	0
$724$	261.917	0.361764
$725$	95.7883i	0.132122i
$726$	0	0
$727$	−755.214	−1.03881	−0.519404	−	0.854529i	$-0.673846\pi$
$727$	−755.214	−1.03881	−0.519404	+	0.854529i	$0.673846\pi$
$728$	3.99213i	0.00548370i
$729$	0	0
$730$	419.796	0.575063
$731$	− 667.517i	− 0.913156i
$732$	0	0
$733$	−126.122	−0.172062	−0.0860311	−	0.996292i	$-0.527418\pi$
$733$	−126.122	−0.172062	−0.0860311	+	0.996292i	$0.527418\pi$
$734$	− 582.055i	− 0.792990i
$735$	0	0
$736$	−34.1642	−0.0464187
$737$	− 99.5653i	− 0.135095i
$738$	0	0
$739$	503.837	0.681782	0.340891	−	0.940103i	$-0.389271\pi$
$739$	503.837	0.681782	0.340891	+	0.940103i	$0.389271\pi$
$740$	282.335i	0.381534i
$741$	0	0
$742$	123.398	0.166305
$743$	− 732.660i	− 0.986083i	−0.870006	−	0.493042i	$-0.835885\pi$
$743$	− 732.660i	− 0.986083i	0.870006	−	0.493042i	$-0.164115\pi$
$744$	0	0
$745$	−213.812	−0.286996
$746$	− 971.604i	− 1.30242i
$747$	0	0
$748$	−240.679	−0.321764
$749$	− 92.6068i	− 0.123641i
$750$	0	0
$751$	−116.216	−0.154749	−0.0773744	−	0.997002i	$-0.524654\pi$
$751$	−116.216	−0.154749	−0.0773744	+	0.997002i	$0.524654\pi$
$752$	27.0356i	0.0359515i
$753$	0	0
$754$	−38.4792	−0.0510335
$755$	604.911i	0.801206i
$756$	0	0
$757$	−644.638	−0.851570	−0.425785	−	0.904824i	$-0.640002\pi$
$757$	−644.638	−0.851570	−0.425785	+	0.904824i	$0.640002\pi$
$758$	218.826i	0.288689i
$759$	0	0
$760$	174.369	0.229433
$761$	1059.55i	1.39231i	0.717891	+	0.696156i	$0.245108\pi$
$761$	1059.55i	1.39231i	−0.717891	+	0.696156i	$0.754892\pi$
$762$	0	0
$763$	−21.6200	−0.0283356
$764$	− 67.9994i	− 0.0890045i
$765$	0	0
$766$	230.690	0.301162
$767$	146.437i	0.190922i
$768$	0	0
$769$	−695.512	−0.904437	−0.452219	−	0.891907i	$-0.649367\pi$
$769$	−695.512	−0.904437	−0.452219	+	0.891907i	$0.649367\pi$
$770$	− 15.6698i	− 0.0203504i
$771$	0	0
$772$	366.217	0.474374
$773$	− 931.046i	− 1.20446i	−0.798323	−	0.602229i	$-0.794279\pi$
$773$	− 931.046i	− 1.20446i	0.798323	−	0.602229i	$-0.205721\pi$
$774$	0	0
$775$	−136.022	−0.175512
$776$	6.47049i	0.00833825i
$777$	0	0
$778$	−438.857	−0.564083
$779$	− 1849.96i	− 2.37479i
$780$	0	0
$781$	−274.781	−0.351833
$782$	− 206.133i	− 0.263597i
$783$	0	0
$784$	−192.050	−0.244961
$785$	81.2704i	0.103529i
$786$	0	0
$787$	−429.601	−0.545871	−0.272936	−	0.962032i	$-0.587995\pi$
$787$	−429.601	−0.545871	−0.272936	+	0.962032i	$0.587995\pi$
$788$	− 61.8702i	− 0.0785154i
$789$	0	0
$790$	−125.192	−0.158471
$791$	− 142.352i	− 0.179965i
$792$	0	0
$793$	−102.277	−0.128975
$794$	381.918i	0.481005i
$795$	0	0
$796$	665.745	0.836363
$797$	610.415i	0.765891i	0.923771	+	0.382945i	$0.125090\pi$
$797$	610.415i	0.765891i	−0.923771	+	0.382945i	$0.874910\pi$
$798$	0	0
$799$	−163.122	−0.204157
$800$	28.2843i	0.0353553i
$801$	0	0
$802$	−857.964	−1.06978
$803$	− 661.927i	− 0.824318i
$804$	0	0
$805$	13.4206	0.0166716
$806$	− 54.6415i	− 0.0677934i
$807$	0	0
$808$	−321.445	−0.397828
$809$	− 304.324i	− 0.376173i	−0.982152	−	0.188087i	$-0.939771\pi$
$809$	− 304.324i	− 0.376173i	0.982152	−	0.188087i	$-0.0602286\pi$
$810$	0	0
$811$	−639.398	−0.788407	−0.394204	−	0.919023i	$-0.628980\pi$
$811$	−639.398	−0.788407	−0.394204	+	0.919023i	$0.628980\pi$
$812$	38.0771i	0.0468930i
$813$	0	0
$814$	445.181	0.546906
$815$	208.683i	0.256053i
$816$	0	0
$817$	762.546	0.933348
$818$	941.721i	1.15125i
$819$	0	0
$820$	300.081	0.365952
$821$	− 1331.48i	− 1.62178i	−0.585196	−	0.810892i	$-0.698982\pi$
$821$	− 1331.48i	− 1.62178i	0.585196	−	0.810892i	$-0.301018\pi$
$822$	0	0
$823$	−891.022	−1.08265	−0.541325	−	0.840813i	$-0.682077\pi$
$823$	−891.022	−1.08265	−0.541325	+	0.840813i	$0.682077\pi$
$824$	− 160.495i	− 0.194775i
$825$	0	0
$826$	144.907	0.175432
$827$	494.513i	0.597960i	0.954259	+	0.298980i	$0.0966465\pi$
$827$	494.513i	0.597960i	−0.954259	+	0.298980i	$0.903354\pi$
$828$	0	0
$829$	−752.153	−0.907302	−0.453651	−	0.891179i	$-0.649879\pi$
$829$	−752.153	−0.907302	−0.453651	+	0.891179i	$0.649879\pi$
$830$	383.229i	0.461722i
$831$	0	0
$832$	−11.3621	−0.0136564
$833$	− 1158.75i	− 1.39106i
$834$	0	0
$835$	−127.294	−0.152448
$836$	− 274.942i	− 0.328878i
$837$	0	0
$838$	305.770	0.364881
$839$	− 506.107i	− 0.603226i	−0.953430	−	0.301613i	$-0.902475\pi$
$839$	− 506.107i	− 0.603226i	0.953430	−	0.301613i	$-0.0975251\pi$
$840$	0	0
$841$	473.984	0.563595
$842$	− 446.190i	− 0.529917i
$843$	0	0
$844$	321.543	0.380976
$845$	− 373.385i	− 0.441876i
$846$	0	0
$847$	95.5398	0.112798
$848$	351.207i	0.414159i
$849$	0	0
$850$	−170.656	−0.200772
$851$	381.281i	0.448039i
$852$	0	0
$853$	22.0081	0.0258008	0.0129004	−	0.999917i	$-0.495894\pi$
$853$	22.0081	0.0258008	0.0129004	+	0.999917i	$0.495894\pi$
$854$	101.208i	0.118511i
$855$	0	0
$856$	263.570	0.307909
$857$	− 122.498i	− 0.142938i	−0.997443	−	0.0714691i	$-0.977231\pi$
$857$	− 122.498i	− 0.142938i	0.997443	−	0.0714691i	$-0.0227687\pi$
$858$	0	0
$859$	1410.03	1.64148	0.820738	−	0.571305i	$-0.193563\pi$
$859$	1410.03	1.64148	0.820738	+	0.571305i	$0.193563\pi$
$860$	123.692i	0.143828i
$861$	0	0
$862$	997.189	1.15683
$863$	674.484i	0.781557i	0.920485	+	0.390778i	$0.127794\pi$
$863$	674.484i	0.781557i	−0.920485	+	0.390778i	$0.872206\pi$
$864$	0	0
$865$	486.658	0.562610
$866$	− 216.000i	− 0.249423i
$867$	0	0
$868$	−54.0705	−0.0622932
$869$	197.401i	0.227159i
$870$	0	0
$871$	28.3599	0.0325602
$872$	− 61.5333i	− 0.0705657i
$873$	0	0
$874$	235.478	0.269426
$875$	− 11.1108i	− 0.0126981i
$876$	0	0
$877$	1370.70	1.56295	0.781473	−	0.623939i	$-0.214469\pi$
$877$	1370.70	1.56295	0.781473	+	0.623939i	$0.214469\pi$
$878$	− 229.434i	− 0.261315i
$879$	0	0
$880$	44.5982	0.0506797
$881$	− 629.262i	− 0.714259i	−0.934055	−	0.357130i	$-0.883755\pi$
$881$	− 629.262i	− 0.714259i	0.934055	−	0.357130i	$-0.116245\pi$
$882$	0	0
$883$	−537.868	−0.609137	−0.304569	−	0.952490i	$-0.598512\pi$
$883$	−537.868	−0.609137	−0.304569	+	0.952490i	$0.598512\pi$
$884$	− 68.5544i	− 0.0775502i
$885$	0	0
$886$	−548.100	−0.618623
$887$	537.138i	0.605568i	0.953059	+	0.302784i	$0.0979160\pi$
$887$	537.138i	0.605568i	−0.953059	+	0.302784i	$0.902084\pi$
$888$	0	0
$889$	−164.026	−0.184506
$890$	234.171i	0.263113i
$891$	0	0
$892$	−782.803	−0.877582
$893$	− 186.344i	− 0.208672i
$894$	0	0
$895$	−125.810	−0.140570
$896$	11.2434i	0.0125484i
$897$	0	0
$898$	157.013	0.174847
$899$	− 521.172i	− 0.579725i
$900$	0	0
$901$	−2119.04	−2.35188
$902$	− 473.163i	− 0.524571i
$903$	0	0
$904$	405.152	0.448177
$905$	− 292.832i	− 0.323572i
$906$	0	0
$907$	498.885	0.550039	0.275019	−	0.961439i	$-0.411316\pi$
$907$	498.885	0.550039	0.275019	+	0.961439i	$0.411316\pi$
$908$	681.398i	0.750439i
$909$	0	0
$910$	4.46334	0.00490477
$911$	− 1132.40i	− 1.24303i	−0.783402	−	0.621515i	$-0.786518\pi$
$911$	− 1132.40i	− 1.24303i	0.783402	−	0.621515i	$-0.213482\pi$
$912$	0	0
$913$	604.269	0.661850
$914$	422.566i	0.462326i
$915$	0	0
$916$	188.893	0.206215
$917$	52.1412i	0.0568606i
$918$	0	0
$919$	−20.1454	−0.0219210	−0.0109605	−	0.999940i	$-0.503489\pi$
$919$	−20.1454	−0.0219210	−0.0109605	+	0.999940i	$0.503489\pi$
$920$	38.1967i	0.0415182i
$921$	0	0
$922$	293.998	0.318870
$923$	− 78.2680i	− 0.0847974i
$924$	0	0
$925$	315.660	0.341254
$926$	− 1223.47i	− 1.32124i
$927$	0	0
$928$	−108.372	−0.116780
$929$	300.765i	0.323751i	0.986811	+	0.161876i	$0.0517543\pi$
$929$	300.765i	0.323751i	−0.986811	+	0.161876i	$0.948246\pi$
$930$	0	0
$931$	1323.71	1.42182
$932$	− 235.965i	− 0.253181i
$933$	0	0
$934$	1208.76	1.29418
$935$	269.087i	0.287794i
$936$	0	0
$937$	310.906	0.331810	0.165905	−	0.986142i	$-0.446945\pi$
$937$	310.906	0.331810	0.165905	+	0.986142i	$0.446945\pi$
$938$	− 28.0635i	− 0.0299185i
$939$	0	0
$940$	30.2267	0.0321560
$941$	1454.01i	1.54518i	0.634908	+	0.772588i	$0.281038\pi$
$941$	1454.01i	1.54518i	−0.634908	+	0.772588i	$0.718962\pi$
$942$	0	0
$943$	405.247	0.429742
$944$	412.423i	0.436889i
$945$	0	0
$946$	195.035	0.206168
$947$	− 190.859i	− 0.201540i	−0.994910	−	0.100770i	$-0.967869\pi$
$947$	− 190.859i	− 0.201540i	0.994910	−	0.100770i	$-0.0321307\pi$
$948$	0	0
$949$	188.542	0.198674
$950$	− 194.951i	− 0.205211i
$951$	0	0
$952$	−67.8379	−0.0712583
$953$	808.807i	0.848695i	0.905499	+	0.424348i	$0.139497\pi$
$953$	808.807i	0.848695i	−0.905499	+	0.424348i	$0.860503\pi$
$954$	0	0
$955$	−76.0257	−0.0796081
$956$	738.226i	0.772203i
$957$	0	0
$958$	−430.117	−0.448974
$959$	− 119.625i	− 0.124739i
$960$	0	0
$961$	−220.922	−0.229888
$962$	126.804i	0.131813i
$963$	0	0
$964$	751.261	0.779316
$965$	− 409.443i	− 0.424293i
$966$	0	0
$967$	427.562	0.442153	0.221076	−	0.975257i	$-0.429043\pi$
$967$	427.562	0.442153	0.221076	+	0.975257i	$0.429043\pi$
$968$	271.918i	0.280907i
$969$	0	0
$970$	7.23422	0.00745796
$971$	932.208i	0.960050i	0.877255	+	0.480025i	$0.159372\pi$
$971$	932.208i	0.960050i	−0.877255	+	0.480025i	$0.840628\pi$
$972$	0	0
$973$	−65.9856	−0.0678166
$974$	− 1283.23i	− 1.31748i
$975$	0	0
$976$	−288.051	−0.295135
$977$	− 203.421i	− 0.208209i	−0.994566	−	0.104105i	$-0.966802\pi$
$977$	− 203.421i	− 0.208209i	0.994566	−	0.104105i	$-0.0331977\pi$
$978$	0	0
$979$	369.236	0.377157
$980$	214.718i	0.219100i
$981$	0	0
$982$	92.5557	0.0942522
$983$	− 1434.02i	− 1.45882i	−0.684074	−	0.729412i	$-0.739794\pi$
$983$	− 1434.02i	− 1.45882i	0.684074	−	0.729412i	$-0.260206\pi$
$984$	0	0
$985$	−69.1729	−0.0702263
$986$	− 653.874i	− 0.663158i
$987$	0	0
$988$	78.3139	0.0792650
$989$	167.041i	0.168898i
$990$	0	0
$991$	−983.596	−0.992528	−0.496264	−	0.868172i	$-0.665295\pi$
$991$	−983.596	−0.992528	−0.496264	+	0.868172i	$0.665295\pi$
$992$	− 153.891i	− 0.155132i
$993$	0	0
$994$	−77.4500	−0.0779175
$995$	− 744.326i	− 0.748066i
$996$	0	0
$997$	−991.171	−0.994154	−0.497077	−	0.867707i	$-0.665593\pi$
$997$	−991.171	−0.994154	−0.497077	+	0.867707i	$0.665593\pi$
$998$	− 1014.39i	− 1.01643i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.d.c.161.7		16
3.2	odd	2	inner	810.3.d.c.161.11		16
9.2	odd	6		90.3.h.a.41.7	yes	16
9.4	even	3		90.3.h.a.11.7	✓	16
9.5	odd	6		270.3.h.a.251.1		16
9.7	even	3		270.3.h.a.71.1		16
36.7	odd	6		2160.3.bs.d.881.3		16
36.11	even	6		720.3.bs.d.401.2		16
36.23	even	6		2160.3.bs.d.1601.3		16
36.31	odd	6		720.3.bs.d.641.2		16
45.2	even	12		450.3.k.c.149.11		32
45.4	even	6		450.3.i.g.101.2		16
45.7	odd	12		1350.3.k.b.449.6		32
45.13	odd	12		450.3.k.c.299.11		32
45.14	odd	6		1350.3.i.g.251.7		16
45.22	odd	12		450.3.k.c.299.6		32
45.23	even	12		1350.3.k.b.899.6		32
45.29	odd	6		450.3.i.g.401.2		16
45.32	even	12		1350.3.k.b.899.11		32
45.34	even	6		1350.3.i.g.1151.7		16
45.38	even	12		450.3.k.c.149.6		32
45.43	odd	12		1350.3.k.b.449.11		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.h.a.11.7	✓	16	9.4	even	3
90.3.h.a.41.7	yes	16	9.2	odd	6
270.3.h.a.71.1		16	9.7	even	3
270.3.h.a.251.1		16	9.5	odd	6
450.3.i.g.101.2		16	45.4	even	6
450.3.i.g.401.2		16	45.29	odd	6
450.3.k.c.149.6		32	45.38	even	12
450.3.k.c.149.11		32	45.2	even	12
450.3.k.c.299.6		32	45.22	odd	12
450.3.k.c.299.11		32	45.13	odd	12
720.3.bs.d.401.2		16	36.11	even	6
720.3.bs.d.641.2		16	36.31	odd	6
810.3.d.c.161.7		16	1.1	even	1	trivial
810.3.d.c.161.11		16	3.2	odd	2	inner
1350.3.i.g.251.7		16	45.14	odd	6
1350.3.i.g.1151.7		16	45.34	even	6
1350.3.k.b.449.6		32	45.7	odd	12
1350.3.k.b.449.11		32	45.43	odd	12
1350.3.k.b.899.6		32	45.23	even	12
1350.3.k.b.899.11		32	45.32	even	12
2160.3.bs.d.881.3		16	36.7	odd	6
2160.3.bs.d.1601.3		16	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.23607i q^{5} +0.993782 q^{7} +2.82843i q^{8} +3.16228 q^{10} -4.98623i q^{11} +1.42026 q^{13} -1.40542i q^{14} +4.00000 q^{16} +24.1344i q^{17} -27.5702 q^{19} -4.47214i q^{20} -7.05159 q^{22} -6.03943i q^{23} -5.00000 q^{25} -2.00856i q^{26} -1.98756 q^{28} -19.1577i q^{29} +27.2044 q^{31} -5.65685i q^{32} +34.1312 q^{34} +2.22216i q^{35} -63.1320 q^{37} +38.9901i q^{38} -6.32456 q^{40} +67.1001i q^{41} -27.6583 q^{43} +9.97245i q^{44} -8.54104 q^{46} +6.75889i q^{47} -48.0124 q^{49} +7.07107i q^{50} -2.84053 q^{52} +87.8018i q^{53} +11.1495 q^{55} +2.81084i q^{56} -27.0930 q^{58} +103.106i q^{59} -72.0128 q^{61} -38.4728i q^{62} -8.00000 q^{64} +3.17581i q^{65} +19.9681 q^{67} -48.2688i q^{68} +3.14262 q^{70} -55.1081i q^{71} +132.751 q^{73} +89.2822i q^{74} +55.1404 q^{76} -4.95522i q^{77} -39.5893 q^{79} +8.94427i q^{80} +94.8939 q^{82} +121.188i q^{83} -53.9661 q^{85} +39.1148i q^{86} +14.1032 q^{88} +74.0513i q^{89} +1.41143 q^{91} +12.0789i q^{92} +9.55851 q^{94} -61.6488i q^{95} +2.28766 q^{97} +67.8998i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 8 q^{7} - 40 q^{13} + 64 q^{16} + 80 q^{19} - 48 q^{22} - 80 q^{25} - 16 q^{28} + 32 q^{31} + 96 q^{34} - 88 q^{37} - 184 q^{43} + 24 q^{46} + 168 q^{49} + 80 q^{52} + 152 q^{61} - 128 q^{64}+ \cdots + 32 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 8 * q^7 - 40 * q^13 + 64 * q^16 + 80 * q^19 - 48 * q^22 - 80 * q^25 - 16 * q^28 + 32 * q^31 + 96 * q^34 - 88 * q^37 - 184 * q^43 + 24 * q^46 + 168 * q^49 + 80 * q^52 + 152 * q^61 - 128 * q^64 - 112 * q^67 - 120 * q^70 + 416 * q^73 - 160 * q^76 - 160 * q^79 - 192 * q^82 + 120 * q^85 + 96 * q^88 - 656 * q^91 + 168 * q^94 + 32 * q^97

Introduction

L-functions

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